Test results from an electronic circuit board indicate that 50% of board failures are caused by assembly defects, 30% by electrical components, and 20% by mechanical defects. Suppose that 10 boards fail independently. Let the random variables X, Y , and Z denote the number of assembly, electrical, and mechanical defects among the 10 boards. 

Calculate the following:

(a) P(X = 5,Y = 3,Z = 2) 

(b) P(X = 8)

(c) P(X = 8 |Y =1) 

(d) P(X ≥ 8 |Y =1)

(e) P(X = 7,Y =1| Z = 2)